This paper is concerned with the analysis of a sixth-order nonlinear parabolic equation whose solutions describe the evolution of the particle density in a quantum fluid. We prove the global-in-time existence of weak nonnegative solutions in two and three space dimensions under periodic boundary conditions. Moreover, we show that these solutions are smooth and classical whenever the particle density is strictly positive, and we prove the long-time convergence to the spatial homogeneous equilibrium at a universal exponential rate. Our analysis strongly uses the Lyapunov property of the entropy functional.
Cite this article
Daniel Matthes, Mario Bukal, Ansgar Jüngel, A multidimensional nonlinear sixth-order quantum diffusion equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 2, pp. 337–365DOI 10.1016/J.ANIHPC.2012.08.003